/*
  This code can be loaded, or copied and pasted, into Magma.
  It will load the data associated to the HMF, including
  the field, level, and Hecke and Atkin-Lehner eigenvalue data.
  At the *bottom* of the file, there is code to recreate the
  Hilbert modular form in Magma, by creating the HMF space
  and cutting out the corresponding Hecke irreducible subspace.
  From there, you can ask for more eigenvalues or modify as desired.
  It is commented out, as this computation may be lengthy.
*/

P<x> := PolynomialRing(Rationals());
g := P![2, 0, -5, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {13, 13, -w^2 + w + 3}>;

primesArray := [
[2, 2, w],
[2, 2, w + 1],
[13, 13, -w^2 + w + 3],
[13, 13, w^2 + w - 3],
[19, 19, -w^3 + 3*w + 1],
[19, 19, -w^3 + 3*w - 1],
[43, 43, -w^2 + w - 1],
[43, 43, w^2 + w + 1],
[49, 7, w^3 + w^2 - 6*w - 3],
[49, 7, w^3 - w^2 - 6*w + 3],
[53, 53, 2*w^3 - w^2 - 9*w + 3],
[53, 53, 2*w^3 + w^2 - 9*w - 3],
[59, 59, w^3 - w^2 - 4*w + 1],
[59, 59, -w^3 - w^2 + 4*w + 1],
[67, 67, 3*w^3 - 13*w + 1],
[67, 67, -w^3 + w^2 + 6*w - 5],
[81, 3, -3],
[83, 83, -2*w^3 - w^2 + 9*w + 7],
[83, 83, 4*w^3 - 18*w - 1],
[89, 89, -2*w^3 + 10*w + 1],
[89, 89, -w^2 + w + 5],
[89, 89, w^2 + w - 5],
[89, 89, 2*w^3 - 10*w + 1],
[101, 101, -2*w^3 + 8*w + 1],
[101, 101, 2*w^3 - 8*w + 1],
[103, 103, w^3 + 2*w^2 - 5*w - 7],
[103, 103, w^3 + w^2 - 6*w - 7],
[103, 103, -w^2 - 3*w + 1],
[103, 103, -w^3 + 2*w^2 + 5*w - 7],
[127, 127, -w^3 + 2*w^2 + 5*w - 5],
[127, 127, w^3 + 3*w^2 - 4*w - 13],
[127, 127, -w^3 + 3*w^2 + 4*w - 13],
[127, 127, w^3 + 2*w^2 - 5*w - 5],
[149, 149, -2*w^3 - w^2 + 7*w - 1],
[149, 149, 2*w^3 - w^2 - 7*w - 1],
[151, 151, -w^3 + w^2 + 2*w - 3],
[151, 151, 2*w^3 - 10*w + 3],
[151, 151, -2*w^3 + 10*w + 3],
[151, 151, w^3 + w^2 - 2*w - 3],
[157, 157, w^3 + 2*w^2 - 3*w - 1],
[157, 157, -w^3 + 2*w^2 + 3*w - 1],
[169, 13, 2*w^2 - 3],
[179, 179, 2*w^3 + w^2 - 7*w - 3],
[179, 179, -2*w^3 + w^2 + 7*w - 3],
[229, 229, -w^3 + 2*w^2 + 3*w - 7],
[229, 229, w^3 + 2*w^2 - 3*w - 7],
[251, 251, -2*w^3 + 2*w^2 + 8*w - 9],
[251, 251, -2*w^3 - 2*w^2 + 8*w + 9],
[263, 263, 2*w^3 - w^2 - 3*w + 1],
[263, 263, -2*w^3 + 2*w^2 + 10*w - 5],
[263, 263, -2*w^3 - 2*w^2 + 10*w + 5],
[263, 263, -2*w^3 - w^2 + 3*w + 1],
[289, 17, 2*w^2 - 5],
[293, 293, 4*w^3 - w^2 - 19*w + 7],
[293, 293, -4*w^3 - w^2 + 19*w + 7],
[307, 307, -w^3 + 3*w - 5],
[307, 307, w^3 - 3*w - 5],
[331, 331, 4*w^3 - 2*w^2 - 18*w + 7],
[331, 331, 5*w^3 - 3*w^2 - 22*w + 11],
[349, 349, -3*w^3 + 13*w + 3],
[349, 349, -3*w^3 + 13*w - 3],
[361, 19, 2*w^2 - 11],
[373, 373, w^3 + 2*w^2 + w - 3],
[373, 373, -w^3 + 2*w^2 - w - 3],
[383, 383, -w^3 + 5*w + 5],
[383, 383, -3*w^3 + 2*w^2 + 11*w - 5],
[383, 383, 3*w^3 + 2*w^2 - 11*w - 5],
[383, 383, w^3 - 5*w + 5],
[389, 389, w^3 - 4*w^2 + w + 7],
[389, 389, -2*w^3 - w^2 + 7*w + 7],
[421, 421, -w^3 - w^2 - 2*w + 1],
[421, 421, w^3 - w^2 + 2*w + 1],
[433, 433, 4*w^3 - 2*w^2 - 16*w + 5],
[433, 433, -4*w + 1],
[433, 433, 4*w + 1],
[433, 433, -5*w^3 + 3*w^2 + 24*w - 13],
[443, 443, -w^3 + w^2 + 3],
[443, 443, -4*w^2 + 6*w + 7],
[457, 457, 3*w^3 + w^2 - 12*w - 3],
[457, 457, -2*w^3 + 6*w - 1],
[457, 457, 2*w^3 - 6*w - 1],
[457, 457, 3*w^3 - w^2 - 12*w + 3],
[461, 461, w^3 - 3*w^2 - 2*w + 7],
[461, 461, w^3 + 3*w^2 - 2*w - 7],
[463, 463, -2*w^3 + 3*w^2 + 9*w - 11],
[463, 463, 2*w^3 - w^2 - 5*w + 1],
[463, 463, -2*w^3 - w^2 + 5*w + 1],
[463, 463, -2*w^3 - 3*w^2 + 9*w + 11],
[467, 467, w^3 - 7*w + 1],
[467, 467, -w^3 + 7*w + 1],
[491, 491, 2*w^2 - 2*w - 7],
[491, 491, 2*w^2 + 2*w - 7],
[509, 509, w^3 + 3*w^2 - 4*w - 11],
[509, 509, -w^3 + 3*w^2 + 4*w - 11],
[523, 523, 2*w^3 - w^2 - 11*w + 3],
[523, 523, 2*w^3 + w^2 - 11*w - 3],
[529, 23, 3*w^2 - w - 15],
[529, 23, -3*w^2 - w + 15],
[557, 557, 3*w^2 + w - 13],
[557, 557, -3*w^2 + w + 13],
[563, 563, -3*w^3 + 3*w^2 + 12*w - 7],
[563, 563, 2*w^3 - 2*w^2 - 8*w + 3],
[569, 569, w^3 + 3*w^2 - 6*w - 9],
[569, 569, 4*w^3 - w^2 - 17*w + 1],
[569, 569, -4*w^3 - w^2 + 17*w + 1],
[569, 569, -w^3 + 3*w^2 + 6*w - 9],
[587, 587, -w^3 + w^2 + 2*w - 5],
[587, 587, w^3 + w^2 - 2*w - 5],
[593, 593, 3*w^3 - 2*w^2 - 15*w + 5],
[593, 593, -w^3 - 2*w^2 + 5*w + 1],
[593, 593, w^3 - 2*w^2 - 5*w + 1],
[593, 593, -3*w^3 - 2*w^2 + 15*w + 5],
[599, 599, w^3 + w^2 - 3],
[599, 599, -w^2 - w - 3],
[599, 599, -w^2 + w - 3],
[599, 599, -w^3 + w^2 - 3],
[613, 613, -w^3 + w^2 + 6*w - 9],
[613, 613, w^3 + w^2 - 6*w - 9],
[625, 5, -5],
[659, 659, -2*w^3 + 4*w^2 + 4*w - 1],
[659, 659, 2*w^3 + 4*w^2 - 4*w - 1],
[661, 661, w^3 + 3*w^2 - 6*w - 7],
[661, 661, -2*w^3 + 6*w^2 - 7],
[701, 701, -w^3 - 2*w^2 + 3*w + 9],
[701, 701, -w^3 + 2*w^2 + 3*w - 9],
[733, 733, w^3 + 2*w^2 - 7*w - 1],
[733, 733, -4*w^3 + w^2 + 19*w - 11],
[739, 739, -w^3 - w^2 + 2*w + 9],
[739, 739, -w^3 + w^2 + 2*w - 9],
[757, 757, 3*w^3 - w^2 - 10*w + 5],
[757, 757, -3*w^3 - w^2 + 10*w + 5],
[773, 773, -4*w^3 + 5*w^2 + 19*w - 23],
[773, 773, -7*w^3 + 6*w^2 + 33*w - 29],
[797, 797, 3*w^3 - 13*w + 5],
[797, 797, 3*w^3 - 13*w - 5],
[829, 829, -2*w^3 - w^2 + 9*w - 1],
[829, 829, 2*w^3 - w^2 - 9*w - 1],
[859, 859, 3*w^3 - w^2 - 14*w + 1],
[859, 859, -3*w^3 - w^2 + 14*w + 1],
[863, 863, -5*w^3 + 21*w + 3],
[863, 863, -3*w^3 - w^2 + 12*w - 1],
[863, 863, 3*w^3 - w^2 - 12*w - 1],
[863, 863, -5*w^3 + 21*w - 3],
[883, 883, -2*w^3 + 4*w^2 + 8*w - 17],
[883, 883, 2*w^3 + 4*w^2 - 8*w - 17],
[953, 953, -w^3 - w^2 + 8*w + 3],
[953, 953, 4*w^3 - 2*w^2 - 16*w + 7],
[953, 953, -3*w^3 + w^2 + 16*w - 7],
[953, 953, w^3 - w^2 - 8*w + 3],
[961, 31, -4*w^3 + 2*w^2 + 12*w + 3],
[961, 31, 3*w^3 - w^2 - 10*w - 1],
[971, 971, 3*w^3 - 4*w^2 - 13*w + 15],
[971, 971, -4*w^3 + 3*w^2 + 19*w - 17],
[977, 977, -2*w^3 + w^2 + 9*w + 3],
[977, 977, -4*w^3 + w^2 + 19*w - 9],
[977, 977, 3*w^3 - 15*w + 5],
[977, 977, 2*w^3 + w^2 - 9*w + 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^7 - 5*x^6 + 2*x^5 + 22*x^4 - 27*x^3 - 13*x^2 + 25*x - 4;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-e^6 + 3*e^5 + 4*e^4 - 15*e^3 - e^2 + 15*e - 1, e, -1, e^5 - 2*e^4 - 5*e^3 + 9*e^2 + 4*e - 4, -2*e^5 + 3*e^4 + 12*e^3 - 13*e^2 - 15*e + 10, -e^6 + 5*e^5 + e^4 - 26*e^3 + 11*e^2 + 26*e - 8, 2*e^5 - 3*e^4 - 12*e^3 + 11*e^2 + 19*e - 10, -2*e^6 + 7*e^5 + 5*e^4 - 33*e^3 + 12*e^2 + 24*e - 12, 3*e^6 - 10*e^5 - 11*e^4 + 50*e^3 - e^2 - 47*e + 12, -e^6 + 3*e^5 + 3*e^4 - 11*e^3 + e^2 + 3*e + 2, 6*e^6 - 22*e^5 - 16*e^4 + 107*e^3 - 24*e^2 - 94*e + 22, 2*e^6 - 5*e^5 - 12*e^4 + 28*e^3 + 19*e^2 - 32*e - 6, e^6 - 5*e^5 + 26*e^3 - 19*e^2 - 25*e + 20, e^5 + e^4 - 8*e^3 - 7*e^2 + 10*e + 6, e^6 - 5*e^5 - 2*e^4 + 25*e^3 - e^2 - 24*e - 4, 2*e^6 - 8*e^5 - 2*e^4 + 37*e^3 - 23*e^2 - 33*e + 16, 3*e^6 - 10*e^5 - 11*e^4 + 50*e^3 + e^2 - 47*e - 2, e^6 - e^5 - 9*e^4 + 8*e^3 + 22*e^2 - 18*e - 12, -3*e^6 + 10*e^5 + 10*e^4 - 49*e^3 - e^2 + 49*e + 6, 2*e^6 - 5*e^5 - 7*e^4 + 23*e^3 - 7*e^2 - 25*e + 22, e^6 - 5*e^5 - 4*e^4 + 28*e^3 + 7*e^2 - 33*e - 2, 5*e^6 - 17*e^5 - 13*e^4 + 77*e^3 - 22*e^2 - 57*e + 24, 5*e^6 - 21*e^5 - 11*e^4 + 107*e^3 - 27*e^2 - 102*e + 22, e^6 - e^5 - 6*e^4 + 3*e^3 + 2*e^2 + 2*e + 18, -7*e^6 + 21*e^5 + 25*e^4 - 101*e^3 + 10*e^2 + 89*e - 22, -6*e^6 + 20*e^5 + 23*e^4 - 103*e^3 - 3*e^2 + 103*e - 12, -2*e^6 + 9*e^5 + 2*e^4 - 45*e^3 + 21*e^2 + 41*e - 10, -e^6 + 7*e^5 - e^4 - 37*e^3 + 11*e^2 + 38*e + 6, 8*e^6 - 29*e^5 - 18*e^4 + 136*e^3 - 47*e^2 - 114*e + 38, -2*e^6 + 4*e^5 + 9*e^4 - 14*e^3 - 4*e^2 - 4*e - 4, 7*e^6 - 23*e^5 - 25*e^4 + 116*e^3 - 9*e^2 - 114*e + 28, -7*e^6 + 21*e^5 + 25*e^4 - 101*e^3 + 10*e^2 + 90*e - 20, -e^6 + e^5 + 8*e^4 - 8*e^3 - 13*e^2 + 18*e, -10*e^6 + 33*e^5 + 30*e^4 - 154*e^3 + 28*e^2 + 126*e - 34, -8*e^6 + 24*e^5 + 34*e^4 - 123*e^3 - 18*e^2 + 125*e, 8*e^6 - 27*e^5 - 21*e^4 + 123*e^3 - 38*e^2 - 92*e + 40, -7*e^6 + 20*e^5 + 27*e^4 - 92*e^3 - 2*e^2 + 75*e - 26, 8*e^6 - 29*e^5 - 25*e^4 + 146*e^3 - 18*e^2 - 141*e + 28, -3*e^6 + 12*e^5 + 6*e^4 - 54*e^3 + 14*e^2 + 35*e - 4, -9*e^6 + 29*e^5 + 30*e^4 - 140*e^3 + 20*e^2 + 123*e - 38, -6*e^6 + 21*e^5 + 14*e^4 - 99*e^3 + 36*e^2 + 89*e - 36, e^6 - 5*e^5 - 2*e^4 + 30*e^3 - 9*e^2 - 43*e + 8, -5*e^6 + 16*e^5 + 21*e^4 - 79*e^3 - 13*e^2 + 68*e + 2, 2*e^6 - 9*e^5 - 7*e^4 + 48*e^3 + 6*e^2 - 47*e - 6, 4*e^6 - 16*e^5 - 4*e^4 + 74*e^3 - 41*e^2 - 65*e + 22, -2*e^6 + 15*e^5 - 4*e^4 - 79*e^3 + 36*e^2 + 77*e - 6, 8*e^6 - 31*e^5 - 18*e^4 + 150*e^3 - 46*e^2 - 123*e + 32, 8*e^6 - 19*e^5 - 40*e^4 + 87*e^3 + 45*e^2 - 66*e - 20, 4*e^6 - 13*e^5 - 14*e^4 + 62*e^3 - 7*e^2 - 46*e + 14, 7*e^6 - 31*e^5 - 9*e^4 + 151*e^3 - 63*e^2 - 131*e + 40, 7*e^6 - 24*e^5 - 26*e^4 + 118*e^3 + 14*e^2 - 112*e - 10, 3*e^6 - 9*e^5 - 12*e^4 + 44*e^3 + 3*e^2 - 37*e + 6, 9*e^6 - 28*e^5 - 28*e^4 + 126*e^3 - 18*e^2 - 101*e + 18, 2*e^6 - 5*e^5 - 11*e^4 + 25*e^3 + 18*e^2 - 29*e - 4, 12*e^6 - 37*e^5 - 45*e^4 + 180*e^3 - 5*e^2 - 159*e + 40, -7*e^6 + 24*e^5 + 27*e^4 - 127*e^3 - 10*e^2 + 132*e + 4, 10*e^6 - 36*e^5 - 27*e^4 + 173*e^3 - 42*e^2 - 148*e + 34, 12*e^6 - 37*e^5 - 46*e^4 + 176*e^3 + 10*e^2 - 147*e + 12, 6*e^6 - 28*e^5 - 10*e^4 + 151*e^3 - 55*e^2 - 162*e + 56, 7*e^6 - 23*e^5 - 21*e^4 + 107*e^3 - 21*e^2 - 96*e + 34, 6*e^6 - 22*e^5 - 13*e^4 + 104*e^3 - 37*e^2 - 91*e + 26, -4*e^6 + 15*e^5 + 11*e^4 - 79*e^3 + 20*e^2 + 89*e - 34, -7*e^6 + 17*e^5 + 30*e^4 - 74*e^3 - 15*e^2 + 54*e - 2, 9*e^6 - 35*e^5 - 19*e^4 + 169*e^3 - 61*e^2 - 144*e + 58, -10*e^6 + 38*e^5 + 22*e^4 - 182*e^3 + 57*e^2 + 155*e - 36, 5*e^6 - 11*e^5 - 18*e^4 + 37*e^3 - 7*e^2 - 6*e + 22, 2*e^6 - 5*e^5 - 8*e^4 + 22*e^3 + 3*e^2 - 8*e - 4, 5*e^6 - 17*e^5 - 17*e^4 + 83*e^3 - 69*e - 4, 2*e^6 - 4*e^5 - 7*e^4 + 12*e^3 + 2*e^2 + 2*e - 10, -e^6 - 4*e^5 + 10*e^4 + 31*e^3 - 17*e^2 - 56*e + 16, -3*e^6 + 6*e^5 + 19*e^4 - 33*e^3 - 34*e^2 + 44*e + 26, -7*e^6 + 24*e^5 + 24*e^4 - 118*e^3 + 7*e^2 + 109*e - 22, 12*e^6 - 42*e^5 - 42*e^4 + 205*e^3 + 10*e^2 - 187*e - 8, 8*e^6 - 22*e^5 - 27*e^4 + 89*e^3 - 10*e^2 - 41*e + 22, 2*e^6 - 9*e^5 + 38*e^3 - 28*e^2 - 25*e + 30, -14*e^6 + 52*e^5 + 33*e^4 - 249*e^3 + 72*e^2 + 222*e - 58, -6*e^6 + 22*e^5 + 16*e^4 - 103*e^3 + 20*e^2 + 89*e - 8, 5*e^6 - 16*e^5 - 8*e^4 + 67*e^3 - 48*e^2 - 48*e + 32, -3*e^6 + 9*e^5 + 2*e^4 - 33*e^3 + 49*e^2 + 9*e - 30, -13*e^6 + 36*e^5 + 56*e^4 - 171*e^3 - 39*e^2 + 150*e + 14, 10*e^6 - 34*e^5 - 34*e^4 + 172*e^3 - 19*e^2 - 169*e + 50, 9*e^6 - 33*e^5 - 24*e^4 + 161*e^3 - 31*e^2 - 157*e + 32, -2*e^5 + e^4 + 18*e^3 - 8*e^2 - 30*e + 4, 7*e^6 - 25*e^5 - 22*e^4 + 127*e^3 - 21*e^2 - 128*e + 38, -e^6 + 6*e^5 + 2*e^4 - 38*e^3 + 5*e^2 + 59*e + 6, 14*e^6 - 49*e^5 - 38*e^4 + 231*e^3 - 61*e^2 - 185*e + 68, 7*e^6 - 18*e^5 - 29*e^4 + 83*e^3 + 7*e^2 - 71*e + 28, -e^5 + 5*e^4 - e^3 - 23*e^2 + 8*e + 8, 5*e^6 - 19*e^5 - 16*e^4 + 93*e^3 + 4*e^2 - 80*e - 12, 4*e^6 - 15*e^5 - 12*e^4 + 75*e^3 - 7*e^2 - 85*e + 14, -9*e^6 + 36*e^5 + 19*e^4 - 177*e^3 + 57*e^2 + 163*e - 52, -4*e^5 + 15*e^4 + 12*e^3 - 58*e^2 - e + 16, 13*e^6 - 43*e^5 - 39*e^4 + 212*e^3 - 48*e^2 - 217*e + 70, -2*e^6 + 3*e^5 + 11*e^4 - 8*e^3 - 23*e^2 + 2*e + 30, 4*e^6 - 23*e^5 + 4*e^4 + 112*e^3 - 67*e^2 - 97*e + 22, -19*e^6 + 62*e^5 + 55*e^4 - 284*e^3 + 60*e^2 + 227*e - 64, -13*e^6 + 40*e^5 + 44*e^4 - 193*e^3 + 25*e^2 + 185*e - 50, 6*e^6 - 25*e^5 - 11*e^4 + 125*e^3 - 45*e^2 - 111*e + 44, -13*e^6 + 49*e^5 + 36*e^4 - 238*e^3 + 29*e^2 + 203*e - 2, 5*e^6 - 16*e^5 - 20*e^4 + 83*e^3 - 89*e + 34, -13*e^6 + 51*e^5 + 29*e^4 - 247*e^3 + 75*e^2 + 201*e - 52, 11*e^6 - 40*e^5 - 37*e^4 + 200*e^3 - 190*e + 12, -6*e^6 + 10*e^5 + 32*e^4 - 43*e^3 - 25*e^2 + 32*e - 6, -11*e^6 + 40*e^5 + 32*e^4 - 200*e^3 + 35*e^2 + 192*e - 26, 4*e^6 - 13*e^5 - 17*e^4 + 71*e^3 + 10*e^2 - 73*e - 18, -3*e^6 + 20*e^5 - 5*e^4 - 101*e^3 + 56*e^2 + 88*e - 26, 5*e^6 - 19*e^5 - 9*e^4 + 94*e^3 - 46*e^2 - 91*e + 32, 2*e^6 - 7*e^5 - 8*e^4 + 37*e^3 + 3*e^2 - 39*e - 4, 6*e^6 - 16*e^5 - 24*e^4 + 73*e^3 + 9*e^2 - 61*e - 20, 9*e^6 - 33*e^5 - 17*e^4 + 153*e^3 - 64*e^2 - 133*e + 42, -3*e^6 + 17*e^5 - 4*e^4 - 84*e^3 + 62*e^2 + 79*e - 50, -12*e^6 + 36*e^5 + 50*e^4 - 182*e^3 - 12*e^2 + 176*e - 26, -4*e^5 + e^4 + 30*e^3 + 4*e^2 - 42*e - 20, 9*e^6 - 36*e^5 - 28*e^4 + 188*e^3 - 9*e^2 - 199*e + 20, -15*e^6 + 51*e^5 + 45*e^4 - 239*e^3 + 42*e^2 + 195*e - 56, 7*e^6 - 32*e^5 - 13*e^4 + 163*e^3 - 48*e^2 - 155*e + 52, -3*e^6 + 9*e^5 + 17*e^4 - 43*e^3 - 38*e^2 + 35*e + 26, 12*e^6 - 39*e^5 - 39*e^4 + 179*e^3 - 27*e^2 - 124*e + 46, -7*e^6 + 13*e^5 + 42*e^4 - 61*e^3 - 66*e^2 + 57*e + 34, 21*e^6 - 67*e^5 - 75*e^4 + 326*e^3 - 14*e^2 - 285*e + 40, 7*e^6 - 22*e^5 - 27*e^4 + 98*e^3 + 16*e^2 - 57*e - 12, -6*e^6 + 5*e^5 + 40*e^4 - 3*e^3 - 77*e^2 - 35*e + 46, -6*e^6 + 21*e^5 + 15*e^4 - 113*e^3 + 55*e^2 + 136*e - 72, 4*e^6 - 10*e^5 - 23*e^4 + 53*e^3 + 37*e^2 - 60*e - 16, 10*e^6 - 32*e^5 - 30*e^4 + 138*e^3 - 19*e^2 - 84*e + 14, -15*e^6 + 62*e^5 + 30*e^4 - 307*e^3 + 93*e^2 + 287*e - 90, 19*e^6 - 61*e^5 - 63*e^4 + 291*e^3 - 35*e^2 - 244*e + 62, 12*e^6 - 42*e^5 - 43*e^4 + 215*e^3 + 11*e^2 - 217*e + 8, 2*e^6 - 9*e^5 - 4*e^4 + 36*e^3 + 11*e^2 - 21*e - 48, 8*e^6 - 26*e^5 - 14*e^4 + 105*e^3 - 64*e^2 - 58*e + 48, 13*e^6 - 42*e^5 - 45*e^4 + 201*e^3 - 9*e^2 - 175*e + 38, 9*e^6 - 29*e^5 - 30*e^4 + 137*e^3 - 18*e^2 - 104*e + 38, -11*e^6 + 27*e^5 + 57*e^4 - 140*e^3 - 62*e^2 + 160*e + 24, e^6 + 6*e^5 - 20*e^4 - 27*e^3 + 48*e^2 + 21*e + 10, 10*e^6 - 34*e^5 - 32*e^4 + 161*e^3 - 5*e^2 - 140*e - 10, -5*e^6 + 13*e^5 + 26*e^4 - 74*e^3 - 23*e^2 + 99*e - 2, -11*e^5 + 19*e^4 + 57*e^3 - 71*e^2 - 46*e + 26, 18*e^6 - 66*e^5 - 50*e^4 + 324*e^3 - 68*e^2 - 282*e + 60, 3*e^6 - 14*e^5 - 2*e^4 + 71*e^3 - 47*e^2 - 76*e + 40, e^6 + 5*e^5 - 19*e^4 - 33*e^3 + 67*e^2 + 58*e - 64, 10*e^6 - 27*e^5 - 36*e^4 + 125*e^3 - 21*e^2 - 108*e + 56, -3*e^6 - 2*e^5 + 34*e^4 + 8*e^3 - 83*e^2 - 4*e + 10, 18*e^6 - 52*e^5 - 68*e^4 + 249*e^3 - 4*e^2 - 234*e + 48, -19*e^6 + 58*e^5 + 71*e^4 - 278*e^3 - e^2 + 243*e - 60, -32*e^6 + 101*e^5 + 112*e^4 - 485*e^3 + 35*e^2 + 424*e - 100, -5*e^6 + 28*e^5 - e^4 - 137*e^3 + 65*e^2 + 127*e - 42, -25*e^6 + 85*e^5 + 78*e^4 - 411*e^3 + 61*e^2 + 370*e - 78, -15*e^6 + 48*e^5 + 58*e^4 - 241*e^3 - 16*e^2 + 240*e - 12, -7*e^6 + 30*e^5 + 4*e^4 - 137*e^3 + 90*e^2 + 98*e - 54, 10*e^6 - 48*e^5 - 12*e^4 + 243*e^3 - 89*e^2 - 232*e + 68, -32*e^6 + 105*e^5 + 106*e^4 - 509*e^3 + 58*e^2 + 462*e - 118, -9*e^6 + 21*e^5 + 45*e^4 - 105*e^3 - 33*e^2 + 119*e - 22, 2*e^6 + 3*e^5 - 21*e^4 - 24*e^3 + 42*e^2 + 39*e + 16, -9*e^6 + 32*e^5 + 19*e^4 - 154*e^3 + 75*e^2 + 131*e - 58, 10*e^6 - 44*e^5 - 27*e^4 + 228*e^3 - 15*e^2 - 230*e + 4, e^6 - 10*e^5 + 9*e^4 + 57*e^3 - 59*e^2 - 72*e + 52, -17*e^6 + 67*e^5 + 34*e^4 - 325*e^3 + 120*e^2 + 268*e - 82];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {13,13,-w^2+w+3}>] := 1;

// EXAMPLE:
// pp := Factorization(2*ZF)[1][1];
// heckeEigenvalues[pp];

print "To reconstruct the Hilbert newform f, type
  f, iso := Explode(make_newform());";

function make_newform();
 M := HilbertCuspForms(F, NN);
 S := NewSubspace(M);
 // SetVerbose("ModFrmHil", 1);
 NFD := NewformDecomposition(S);
 newforms := [* Eigenform(U) : U in NFD *];

 if #newforms eq 0 then;
  print "No Hilbert newforms at this level";
  return 0;
 end if;

 print "Testing ", #newforms, " possible newforms";
 newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *];
 print #newforms, " newforms have the correct Hecke field";

 if #newforms eq 0 then;
  print "No Hilbert newform found with the correct Hecke field";
  return 0;
 end if;

 autos := Automorphisms(K);
 xnewforms := [* *];
 for f in newforms do;
  if K eq RationalField() then;
   Append(~xnewforms, [* f, autos[1] *]);
  else;
   flag, iso := IsIsomorphic(K,BaseField(f));
   for a in autos do;
    Append(~xnewforms, [* f, a*iso *]);
   end for;
  end if;
 end for;
 newforms := xnewforms;

 for P in primes do;
  xnewforms := [* *];
  for f_iso in newforms do;
   f, iso := Explode(f_iso);
   if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then;
    Append(~xnewforms, f_iso);
   end if;
  end for;
  newforms := xnewforms;
  if #newforms eq 0 then;
   print "No Hilbert newform found which matches the Hecke eigenvalues";
   return 0;
  else if #newforms eq 1 then;
   print "success: unique match";
   return newforms[1];
  end if;
  end if;
 end for;
 print #newforms, "Hilbert newforms found which match the Hecke eigenvalues";
 return newforms[1];

end function;